We prove that any H-minor-free graph, for a fixed graph H,
of treewidth w has an
Ω(w) × Ω(w) grid graph as a minor.
Thus grid minors suffice to certify that H-minor-free graphs have large
treewidth, up to constant factors. This strong relationship was previously
known for the special cases of planar graphs and bounded-genus graphs, and is
known not to hold for general graphs. The approach of this paper can be
viewed more generally as a framework for extending combinatorial results on
planar graphs to hold on H-minor-free graphs for any
fixed H. Our result has many combinatorial consequences on
bidimensionality theory, parameter-treewidth bounds, separator theorems, and
bounded local treewidth; each of these combinatorial results has several
algorithmic consequences including subexponential fixed-parameter algorithms
and approximation algorithms.